Explicit solutions of nonlocal reverse-time Hirota-Maxwell-Bloch system

Zh. Myrzakulova; Z. Zakariyeva; K. Suleimenov; U. Uralbekova; K. Yesmakhanova; Zh. Myrzakulova; Z. Zakariyeva; K. Suleimenov; U. Uralbekova; K. Yesmakhanova

doi:10.3934/math.20241666

AIMS Mathematics

2024, Volume 9, Issue 12: 35004-35015. doi: 10.3934/math.20241666

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Research article

Explicit solutions of nonlocal reverse-time Hirota-Maxwell-Bloch system

1.
Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, 13 Kazhymukan, Astana, 010008, Kazakhstan
2.
Faculty of Physics and Mathematics, M. Utemisov West Kazakhstan University, 162 Nazarbayev, Oral, 090000, Kazakhstan
3.
Faculty of Mechanics and Mathematics, al-Farabi Kazakh National University, 71 al-Farabi, Almaty, 050040, Kazakhstan

Received: 01 July 2024 Revised: 26 November 2024 Accepted: 29 November 2024 Published: 16 December 2024
MSC : 35B06, 35C08

In this paper, we investigate the nonlocal reverse-time Hirota-Maxwell-Bloch system, focusing on its soliton solutions using the Darboux transformation method. By deriving the Darboux transformation for this system, we obtained explicit expressions for the new potentials $ q', p' $, and $ \eta' $ in both the defocusing ($ \kappa = 1 $) and focusing ($ \kappa = -1 $) cases. Our analysis reveals significant differences in soliton behavior depending on the value of $ \kappa $, with the defocusing case producing wide, smooth solitons and the focusing case yielding narrow, highly localized solitons. These results provide a deeper understanding of soliton dynamics in nonlocal integrable systems and lay the groundwork for future studies on the influence of nonlocality in integrable models.
- nonlocal reverse-time Hirota and Maxwell-Bloch system,
- PT-symmetric,
- Darboux transformation,
- explicit solution,
- Lax pair
Citation: Zh. Myrzakulova, Z. Zakariyeva, K. Suleimenov, U. Uralbekova, K. Yesmakhanova. Explicit solutions of nonlocal reverse-time Hirota-Maxwell-Bloch system[J]. AIMS Mathematics, 2024, 9(12): 35004-35015. doi: 10.3934/math.20241666

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Abstract

In this paper, we investigate the nonlocal reverse-time Hirota-Maxwell-Bloch system, focusing on its soliton solutions using the Darboux transformation method. By deriving the Darboux transformation for this system, we obtained explicit expressions for the new potentials $ q', p' $, and $ \eta' $ in both the defocusing ($ \kappa = 1 $) and focusing ($ \kappa = -1 $) cases. Our analysis reveals significant differences in soliton behavior depending on the value of $ \kappa $, with the defocusing case producing wide, smooth solitons and the focusing case yielding narrow, highly localized solitons. These results provide a deeper understanding of soliton dynamics in nonlocal integrable systems and lay the groundwork for future studies on the influence of nonlocality in integrable models.

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